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Hooke's Law Calculator

Force equals negative spring constant times displacement from equilibrium

Solution

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Hooke’s Law (Force)

Hooke’s Law states that the restoring force of a spring is proportional to its displacement from equilibrium. The spring constant k measures stiffness — a higher k means a stiffer spring. You can solve for force, distance, equilibrium position, or spring constant.

F = −k(x − x₀)

Spring Potential Energy

The elastic potential energy stored in a spring equals one-half times the spring constant times the displacement squared. This energy is released as kinetic energy when the spring returns to its natural length.

U = ½kx²

How It Works

Hooke’s Law states that the force needed to stretch or compress a spring is proportional to the displacement: F = −k(x − x₀). The spring constant k measures stiffness — a higher k means a stiffer spring. The stored potential energy is U = ½kx². This law applies to any elastic material within its proportional limit, including rubber bands, metal beams, and biological tissues.

Example Problem

A spring with k = 200 N/m is stretched 0.15 m from its natural length. What force is required?

  1. F = k × x = 200 × 0.15 = 30 N
  2. Potential energy stored: U = ½ × 200 × 0.15² = 2.25 J

When to Use Each Variable

  • Solve for Force — when you know the spring constant and displacement, e.g., calculating the restoring force of a compressed suspension spring.
  • Solve for Distance — when you know the force and spring constant, e.g., finding how far a spring stretches under a known load.
  • Solve for Spring Constant — when you know the force and displacement, e.g., determining stiffness from a load-deflection test.
  • Solve for Potential Energy — when you know the spring constant and displacement, e.g., calculating the energy stored in a drawn bowstring.

Key Concepts

Hooke's Law states that the restoring force of an elastic material is proportional to its displacement from equilibrium: F = -k(x - x₀). The spring constant k (in N/m) measures stiffness — higher k means more force is needed for the same displacement. The elastic potential energy stored in a spring is U = ½kx². Hooke's Law applies only within the elastic limit; beyond that point, permanent deformation occurs.

Applications

  • Automotive engineering: designing suspension springs for ride comfort and load-bearing capacity
  • Mechanical watches: calibrating mainsprings and balance springs for accurate timekeeping
  • Civil engineering: analyzing deflection in beams and structural members within the elastic range
  • Medical devices: designing spring-loaded syringes, retractors, and prosthetic joints

Common Mistakes

  • Applying Hooke's Law beyond the elastic limit — once a material yields, the linear relationship breaks down
  • Forgetting the sign convention — the restoring force opposes the displacement direction (negative sign)
  • Confusing displacement from equilibrium with total length — x in the formula is the change from natural length, not the stretched length
  • Using the wrong units for k — spring constant must be in force per length (e.g., N/m), not force alone

Frequently Asked Questions

What is a spring constant?

The spring constant k (in N/m) measures how stiff a spring is. A k of 500 N/m means you need 500 N of force to stretch the spring by 1 meter.

Does Hooke’s Law work for all materials?

Only within the elastic (proportional) limit. Beyond that point the material deforms permanently and the linear relationship breaks down. Metals, for example, follow Hooke’s Law up to their yield strength.

How is spring potential energy used?

Spring potential energy (U = ½kx²) is stored when a spring is compressed or stretched and released as kinetic energy. It powers mechanical watches, vehicle suspension systems, and spring-loaded mechanisms.

Reference: Tipler, Paul A. 1995. Physics For Scientists and Engineers. Worth Publishers. 3rd ed.

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